Varieties of algebras with nontrivial identities on subalgebra lattices (Q5937513)

For a variety \textbf{V}, let \(\text{Sub }{\mathbf V}\) be the class of subalgebra lattices of all members of \textbf{V}. The author examines varieties of groups, semigroups, associative or Lie rings \textbf{V} such that \(\text{Sub }{\mathbf V}\) satisfies some non-trivial lattice identity. The main results of the article are Theorems 1-4. Theorem 1 says that for a semigroup variety \textbf{V} the following are equivalent: (i) the lattices of \(\text{Sub }{\mathbf V}\) satisfy the same non-trivial identity; (ii) the subsemigroup lattice of any member of \textbf{V} satisfies a non-trivial identity; (iii) \textbf{V} is a periodic variety and consists of nilpotent extensions of completely simple semigroups, and the subgroup lattice of any group from \textbf{V} satisfies a non-trivial identity. Theorem 2 states that if \textbf{V} is a group variety and the lattices of \(\text{Sub }{\mathbf V}\) satisfy a non-trivial identity then \textbf{V} is solvable. In Theorem 3 the author gives a countable series of lattice identities \(u_n\leq v_n\) with the following property: if \textbf{V} is a variety of semigroups with a distinguished zero, groups, associative or Lie rings, then, for any \(n\), \(\text{Sub }{\mathbf V}\) satisfies the identity \(u_n\leq v_n\) if and only if \textbf{V} satisfies the identity \([x_1,\dots,x_{n+1}]=1\) in the case of groups and \(x_1 \cdots x_{n+1}=0\) in the other cases. Theorem 4 is inverse, in a sense, to Theorem 3 in the case of associative ring varieties. Namely, Theorem 4 states that, for a variety of associative rings \textbf{V}, the following are equivalent: (i) the lattices of \(\text{Sub }{\mathbf V}\) satisfy a non-trivial identity; (ii) the lattices of \(\text{Sub }{\mathbf V}\) satisfy the identity \(u_n\leq v_n\) for a suitable \(n\); (iii) \textbf{V} is a nilpotent variety. No proofs are included.